Double Angle Formulae (Leaving Cert Mathematics): Revision Notes

Double Angle Formulae

Double angle formulae are trigonometric identities that express the trigonometric functions of double angles (i.e., 2θ) in terms of the trigonometric functions of the original angle θ. These formulae are very useful in simplifying expressions, solving trigonometric equations, and analysing oscillatory motion in physics.

Sine of a Double Angle:

The sine of a double angle is given by:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

• This formula is derived from the sine sum identity $\sin(A + B) = \sin A \cos B + \cos A \sin B$ by setting $A = B = \theta.$

Cosine of a Double Angle:

The cosine of a double angle has three equivalent forms:

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

Using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$, the above can also be written as:

$\cos(2\theta) = 2\cos^2(\theta) - 1$ or $\cos(2\theta) = 1 - 2\sin^2(\theta)$

• These forms are useful depending on whether you know or want to simplify the expression in terms of sine or cosine.

Tangent of a Double Angle:

The tangent of a double angle is given by:

$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

• This formula is derived from the tangent sum identity $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} by\ setting\ A = B = \theta.$

Example Problems Using Double Angle Formulae:

Applications of Double Angle Formulae:

• Simplifying Trigonometric Expressions: Double angle formulae are often used to rewrite trigonometric expressions in a simpler form, especially when dealing with powers of trigonometric functions.
• Solving Trigonometric Equations: These identities are crucial for solving equations where angles are involved in double forms.
• Calculus: In integration and differentiation, double angle identities help simplify integrals or derivatives involving trigonometric functions.
• Physics: Double angle identities are used in wave mechanics, signal processing, and oscillations.

Summary:

• Double angle formulae are key trigonometric identities that simplify the analysis of trigonometric functions when dealing with angles that are multiples of other angles.
• The formulae for sine, cosine, and tangent of double angles are essential tools for simplifying expressions, solving equations, and applying trigonometric concepts in calculus and physics.
• Mastery of these identities enhances your ability to tackle more complex problems in trigonometry and related fields.

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Double Angle Formulae

The compound angle formulae can be used to derive double angle formulae.

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Solve each equation for $x$ in the interval $0 \leq x \leq 360^\circ$.

Give your answers to $1$ decimal place where appropriate.

a) $\cos(2x) + \cos x = 0$

1. $\cos 2x = \cos^2 x - \sin^2 x \equiv 2\cos^2 x - 1$
2. $2\cos^2 x - 1 + \cos x = 0 \Rightarrow 2\cos^2 x + \cos x - 1 = 0$
3. Let $\cos x = \frac{1}{2}$

$$x = \arccos\left(\frac{1}{2}\right) = 60, 300$$ $$\cos x = -1 \Rightarrow x = \arccos(-1) = 180$$

Thus, the solutions are:

$$x = 60, 180, 300$$

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Solve each equation for x in the interval $0 \leq x \leq 2\pi^c$.

Give your answers to 1 decimal place where appropriate.

b) $\sin 2x + \cos x = 0$

$$2\sin x \cos x + \cos x = 0$$

Expand $\sin 2x$:

• Bad Method:

$$2\sin x \cos x + 1 = 0 \quad \text{(Divide by } \cos x\text{)}$$ $$2\sin x = -1$$ $$\sin x = -\frac{1}{2}$$ $$x = -\frac{1}{6}\pi, \pi + \frac{1}{6}\pi, 2\pi - \frac{1}{6}\pi$$

Graph drawn below the calculations to show solutions.

$$x = \frac{7\pi}{6}, \frac{11\pi}{6}$$ $$\text{Solution missing for } \cos x = 0$$

• Good Method:

$$\cos x (2\sin x + 1) = 0$$ $$2\sin x + 1 = 0, \quad \cos x = 0$$ $$2\sin x = -1$$ $$\sin x = -\frac{1}{2}$$ $$x = -\frac{1}{6}\pi, \pi + \frac{1}{6}\pi, 2\pi - \frac{1}{6}\pi$$

Graph drawn below the calculations to show solutions.

$$x = \frac{7\pi}{6}, \frac{11\pi}{6}$$ $$\cos x = 0 \Rightarrow x = \frac{\pi}{2}, \frac{3\pi}{2}$$

Graph drawn below the calculations to show solutions.

$$\boxed {x = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6}}$$

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